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Runcinated 7-simplexes
| Orthogonal projections in A7 Coxeter plane |
|---|
In seven-dimensional geometry, a runcinated 7-simplex is a convex uniform 7-polytope with 3rd order truncations (runcination) of the regular 7-simplex.
There are 8 unique runcinations of the 7-simplex with permutations of truncations, and cantellations.
Runcinated 7-simplex
| Runcinated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
Alternate names
- Small prismated octaexon (acronym: spo) (Jonathan Bowers)
Coordinates
The vertices of the runcinated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,1,2). This construction is based on facets of the runcinated 8-orthoplex.
Images
Biruncinated 7-simplex
| Biruncinated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
Alternate names
- Small biprismated octaexon (sibpo) (Jonathan Bowers)
Coordinates
The vertices of the biruncinated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,1,2,2). This construction is based on facets of the biruncinated 8-orthoplex.
Images
Runcitruncated 7-simplex
| runcitruncated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
Alternate names
- Prismatotruncated octaexon (acronym: patto) (Jonathan Bowers)
Coordinates
The vertices of the runcitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,2,3). This construction is based on facets of the runcitruncated 8-orthoplex.
Images
Biruncitruncated 7-simplex
| Biruncitruncated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
Alternate names
- Biprismatotruncated octaexon (acronym: bipto) (Jonathan Bowers)
Coordinates
The vertices of the biruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,3,3). This construction is based on facets of the biruncitruncated 8-orthoplex.
Images
Runcicantellated 7-simplex
| runcicantellated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
Alternate names
- Prismatorhombated octaexon (acronym: paro) (Jonathan Bowers)
Coordinates
The vertices of the runcicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,2,3). This construction is based on facets of the runcicantellated 8-orthoplex.
Images
Biruncicantellated 7-simplex
| biruncicantellated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
Alternate names
- Biprismatorhombated octaexon (acronym: bipro) (Jonathan Bowers)
Coordinates
The vertices of the biruncicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,2,3,3). This construction is based on facets of the biruncicantellated 8-orthoplex.
Images
Runcicantitruncated 7-simplex
| runcicantitruncated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
Alternate names
- Great prismated octaexon (acronym: gapo) (Jonathan Bowers)
Coordinates
The vertices of the runcicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,3,4). This construction is based on facets of the runcicantitruncated 8-orthoplex.
Images
Biruncicantitruncated 7-simplex
| biruncicantitruncated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter group |
| Properties |
Alternate names
- Great biprismated octaexon (acronym: gibpo) (Jonathan Bowers)
Coordinates
The vertices of the biruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,3,4,4). This construction is based on facets of the biruncicantitruncated 8-orthoplex.
Images
Notes
References
- H.S.M. Coxeter:
- H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
- Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, wiley.com,
- (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
- (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
- Norman Johnson Uniform Polytopes, Manuscript (1991)
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
- x3o3o3x3o3o3o - spo, o3x3o3o3x3o3o - sibpo, x3x3o3x3o3o3o - patto, o3x3x3o3x3o3o - bipto, x3o3x3x3o3o3o - paro, x3x3x3x3o3o3o - gapo, o3x3x3x3x3o3o- gibpo
References
- Klitzing, (x3o3o3x3o3o3o - spo)
- Klitzing, (o3x3o3o3x3o3o - sibpo)
- Klitzing, (x3x3o3x3o3o3o - patto)
- Klitzing, (o3x3x3o3x3o3o - bipto)
- Klitzing, (x3o3x3x3o3o3o - paro)
- Klitzing, (x3x3x3x3o3o3o - gapo)
- Klitzing, (o3x3x3x3x3o3o- gibpo)
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